Simple groups, generation and probabilistic methods

It is well known that every finite simple group can be generated by two elements and this leads naturally to a wide range of problems that have been the focus of intensive research in recent years, such as random generation, (2,3)-generation and so on. In this talk I will discuss some recent progress on similar problems for subgroups of simple groups, with applications to primitive permutation groups and the study of subgroup growth (this is joint work with Liebeck and Shalev). I will also recall the notion of the spread of a finite group and I will explain how probabilistic methods (based on fixed point ratio estimates for simple groups) have been used to shed light on a far reaching conjecture of Breuer, Guralnick and Kantor. Time permitting, I will finish by mentioning some related problems on the generating graph of a finite group.